A294704 -id:A294704 - cp's OEIS frontend

A294757 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^(k^k).

1, 1, 17, 746, 66442, 9843731, 2187951485, 680615166718, 282199710311343, 150389915850565698, 100155578811552469018, 81505577529171038120173, 79580089696277797740768316, 91814299717377746850767747558

Offset: 0

Seiichi Manyama, Nov 08 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..214

Column k=1 of A294756.

Cf. A294704, A294773.

N=20; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^k*x^k)^k^k))

sd(n) = sumdiv(n, d, d^(d+n+1));
a(n) = if (n==0, 1, sum(k=1, n, sd(k)*a(n-k))/n); \\ Michel Marcus, Nov 10 2017

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294773(k)*a(n-k) for n > 0.

a(n) ~ n^(2*n). - Vaclav Kotesovec, Nov 08 2017

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -16, 0, 1, -1, -256, -713, 0, 1, -1, -4096, -531185, -64711, 1, 1, -1, -65536, -387416393, -4294405135, -9688521, 0, 1, -1, -1048576, -282429470945, -281474581032631, -95363000655153, -2165724176, 1, 1

Offset: 0

Seiichi Manyama, Nov 07 2017

			Square array begins:
    1,      1,           1,                1, ...
   -1,     -1,          -1,               -1, ...
   -1,    -16,        -256,            -4096, ...
    0,   -713,     -531185,       -387416393, ...
    0, -64711, -4294405135, -281474581032631, ...

Seiichi Manyama, Antidiagonals n = 0..38, flattened

Columns k=0..1 give A010815, A294704.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294583, A294653, A294756.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*(d+j))) * A(n-j,k) for n > 0.

A294773 a(n) = Sum_{d|n} d^(d+n+1).

Original entry on oeis.org

1, 33, 2188, 262273, 48828126, 13060753578, 4747561509944, 2251799880796161, 1350851717674586413, 1000000000152587898818, 895430243255237372246532, 953962166441299506564257602, 1192533292512492016559195008118

Offset: 1

Seiichi Manyama, Nov 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..214

Cf. A294645, A294704, A294757.

a[n_] := DivisorSum[n, #^(# + n + 1) &]; Array[a, 13] (* Amiram Eldar, Oct 04 2023 *)

PARI
```
{a(n) = sumdiv(n, d, d^(d+n+1))}
```

cp's OEIS Frontend

A294757 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^(k^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294773 a(n) = Sum_{d|n} d^(d+n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

cp's OEIS Frontend

A294757 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^(k^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294773 a(n) = Sum_{d|n} d^(d+n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A294699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j(k*j) in powers of x.